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Home Applications of the Integral Improper Integrals Problems  
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In Problems 136, test the improper integral for convergence and evaluate when possible. 37 Show that if r is a rational number, the improper integral converges when r < 1 and diverges when r > 1. 38 Show that if r is rational, the improper integral converges when r > 1 and diverges when r < 1. 39 Find the area of the region under the curve y = 4x^{2} from x = 1 to x = x. 40 Find the area of the region under the curve from x = ½ to x = 1. 41 Find the area of the region between the curves y = x^{1/4} and y = x^{1/2} from x = 0 to x = 1. 42 Find the area of the region between the curves y = x^{3} and y = x^{2}, 1 ≤ x < x. 43 Find the volume of the solid generated by rotating the curve y = l/x, 1 ≤ x < x, about (a) the xaxis, (b) the yaxis. 44 Find the volume of the solid generated by rotating the curve y = x^{1/3}, 0 < x ≤ 1, about (a) the xaxis, (b) the yaxis. 45 Find the volume of the solid generated by rotating the curve y = x^{3/2}, 0 < v ≤ 4, about (a) the xaxis, (b) the yaxis. 46 Find the volume generated by rotating the curve y = 4x^{3}, ∞ < ∞ ≤ 2, about (a) the xaxis, (b) the yaxis. 47 Find the length of the curve from x = 0 to x = 1. 48 Find the length of the curve y = ¾x^{1/3}  3/5 x^{5/3} from x = 0 to x = 1. 49 Find the surface area generated when the curve y = √x  ⅓x√x, 0 ≤ x ≤ 1, is rotated about (a) the xaxis, (b) the yaxis. 50 Do the same for the curve y = ¾x^{1/3}  3/5 x^{5/3},0 ≤ x ≤ 1. 51 (a) Find the surface area generated by rotating the curve y = √x, 0 ≤ x ≤ 1, about the xaxis. (b) Set up an integral for the area generated about the yaxis. 52 Find the surface area generated by rotating the curve y = x^{2/3}, 0 ≤ x ≤ 8, about the xaxis. 53 Find the surface area generated by rotating the curve , 0 ≤ x ≤ a, about (a) the xaxis, (b) the yaxis (0 < a ≤ r). 54 The force of gravity between particles of mass m_{1} and m_{2} is F = gm_{1}m_{2}/s^{2} where s is the distance between them. If m_{1} is held fixed at the origin, find the work done in moving m_{2} from the point (1,0) all the way out the xaxis. H 55 Show that the Rectangle and Addition Properties hold for improper integrals.


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